## PT-Symmetric Optomechanically-Induced Transparency

H. Jing, Z. Geng, S. K. Özdemir, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, F. Nori

Optomechanically-induced transparency (OMIT) and the associated slow-light propagation provide the basis for storing photons in nanofabricated phononic devices. Here we study OMIT in parity-time (PT)-symmetric microresonators with a tunable gain-to-loss ratio. This system features a reversed, non-amplifying transparency: inverted-OMIT. When the gain-to-loss ratio is steered, the system exhibits a transition from the PT-symmetric phase to the broken-PT-symmetric phase. We show that by tuning the pump power at fixed gain-to-loss ratio or the gain-to-loss ratio at fixed pump power, one can switch from slow to fast light and vice versa. Moreover, the presence of PT-phase transition results in the reversal of the pump and gain dependence of transmission rates. These features provide new tools for controlling light propagation using optomechanical devices.

http://arxiv.org/abs/1411.7115
Quantum Physics (quant-ph); Optics (physics.optics)

## Loss-induced suppression and revival of lasing

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, L. Yang

Controlling and reversing the effects of loss are major challenges in optical systems. For lasers losses need to be overcome by a sufficient amount of gain to reach the lasing threshold. We show how to turn losses into gain by steering the parameters of a system to the vicinity of an exceptional point (EP), which occurs when the eigenvalues and the corresponding eigenstates of a system coalesce. In our system of coupled microresonators, EPs are manifested as the loss-induced suppression and revival of lasing. Below a critical value, adding loss annihilates an existing Raman laser. Beyond this critical threshold, lasing recovers despite the increasing loss, in stark contrast to what would be expected from conventional laser theory. Our results exemplify the counterintuitive features of EPs and present an innovative method for reversing the effect of loss.

http://arxiv.org/abs/1410.7474
Optics (physics.optics); Quantum Physics (quant-ph)

## PT-symmetric interpretation of the electromagnetic self-force

Carl M. Bender, Mariagiovanna Gianfreda

In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a charge-conjugate particle and to show that the two-particle system is Hamiltonian. Unfortunately, Englert’s model did not solve the problem of runaway modes, and the corresponding quantum theory had ghost states. It is shown here that Englert’s Hamiltonian is PT symmetric, and that the problems with his model arise because the PT symmetry is broken at both the classical and quantum level. However, by allowing the charged particles to interact and by adjusting the coupling parameters to put the model into an unbroken PT-symmetric region, one eliminates the classical runaway modes and obtains a corresponding quantum system that is ghost free.

http://arxiv.org/abs/1409.3828
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

## Infinitely many inequivalent field theories from one Lagrangian

Carl M. Bender, Daniel W. Hook, Nick E. Mavromatos, Sarben Sarkar

Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field $$\phi$$. In Euclidean space the Lagrangian of such a theory, $$L=\frac{1}{2}(\nabla\phi)^2−ig\phi \exp(ia\phi)$$, is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the $$m$$th energy level in the $$n$$th sector is given by $$E_{m,n}∼(m+1/2)^2a^2/(16n^2)$$.

http://arxiv.org/abs/1408.2432
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

## Giant Optomechanical Enhancement in the Presence of Gain and Loss

H. Jing, Sahin K. Ozdemir, Xin-You Lv, Jing Zhang, F. Nori

The parity-time-symmetric structure was experimentally accessible very recently in coupled optical resonators with which, for normal or non-PT-symmetric cases, a phonon laser device had also been realized. Here we study cavity optomechanics of this system now with tunable gain-loss ratio. We find that nonlinear behaviors emerge for cavity-photon populations around balanced point, resulting giant enhancement of both optical pressure and phonon-lasing action. Potential applications range from enhancing mechanical cooling to designing highly-efficient phonon-laser amplifier.

http://arxiv.org/abs/1403.0657
Quantum Physics (quant-ph); Optics (physics.optics)

## Complex classical motion in potentials with poles and turning points

Carl M. Bender, Daniel W. Hook

Complex trajectories for Hamiltonians of the form H=p^n+V(x) are studied. For n=2 time-reversal symmetry prevents trajectories from crossing. However, for n>2 trajectories may indeed cross, and as a result, the complex trajectories for such Hamiltonians have a rich and elaborate structure. In past work on complex classical trajectories it has been observed that turning points act as attractors; they pull on complex trajectories and make them veer towards the turning point. In this paper it is shown that the poles of V(x) have the opposite effect — they deflect and repel trajectories. Moreover, poles shield and screen the effect of turning points.

http://arxiv.org/abs/1402.3852
Mathematical Physics (math-ph)

## Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities

Bo Peng, Sahin Kaya Ozdemir, Fuchuan Lei, Faraz Monifi, Mariagiovanna Gianfreda, Gui Lu Long, Shanhui Fan, Franco Nori, Carl M. Bender, Lan Yang

Optical systems combining balanced loss and gain profiles provide a unique platform to implement classical analogues of quantum systems described by non-Hermitian parity-time- (PT-) symmetric Hamiltonians and to originate new synthetic materials with novel properties. To date, experimental works on PT-symmetric optical systems have been limited to waveguides in which resonances do not play a role. Here we report the first demonstration of PT-symmetry breaking in optical resonator systems by using two directly coupled on-chip optical whispering-gallery-mode (WGM) microtoroid silica resonators. Gain in one of the resonators is provided by optically pumping Erbium (Er3+) ions embedded in the silica matrix; the other resonator exhibits passive loss. The coupling strength between the resonators is adjusted by using nanopositioning stages to tune their distance. We have observed reciprocal behavior of the PT-symmetric system in the linear regime, as well as a transition to nonreciprocity in the PT symmetry-breaking phase transition due to the significant enhancement of nonlinearity in the broken-symmetry phase. Our results represent a significant advance towards a new generation of synthetic optical systems enabling on-chip manipulation and control of light propagation.

http://arxiv.org/abs/1308.4564
Optics (physics.optics); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Classical Physics (physics.class-ph); Quantum Physics (quant-ph)

## Double-Scaling Limit of the O(N)-Symmetric Anharmonic Oscillator

Carl M. Bender, Sarben Sarkar

In an earlier paper it was argued that the conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative and thus the integral representing the partition function of the critical theory does not exist. In this earlier paper it was shown that for an O(N)-symmetric quantum field theory in zero-dimensional spacetime one can avoid this difficulty if one replaces the original quartic theory by its PT-symmetric analog. In the current paper an O(N)-symmetric quartic quantum field theory in one-dimensional spacetime [that is, O(N)-symmetric quantum mechanics] is studied using the Schroedinger equation. It is shown that the global PT-symmetric formulation of this differential equation provides a consistent way to perform the double-scaling limit of the O(N)-symmetric anharmonic oscillator. The physical nature of the critical behavior is explained by studying the PT-symmetric quantum theory and the corresponding and equivalent Hermitian quantum theory.

http://arxiv.org/abs/1307.4348

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

## The Sign Problem, PT Symmetry and Abelian Lattice Duality

Peter N. Meisinger, Michael C. Ogilvie

Lattice field theories with complex actions are not easily studied using conventional analytic or simulation methods. However, a large class of these models are invariant under CT, where C is charge conjugation and T is time reversal, including models with non-zero chemical potential. For Abelian models in this class, lattice duality maps models with complex actions into dual models with real actions. For extended regions of parameter space, calculable for each model, duality resolves the sign problem for both analytic methods and computer simulations. Explicit duality relations are given for models for spin and gauge models based on Z(N) and U(1) symmetry groups. The dual forms are generalizations of the Z(N) chiral clock model and the lattice Frenkel-Kontorova model, respectively. From these equivalences, rich sets of spatially-modulated phases are found in the strong-coupling region of the original models.

http://arxiv.org/abs/1306.1495
High Energy Physics – Lattice (hep-lat)

## Twofold Transition in PT-Symmetric Coupled Oscillators

Carl M. Bender, Mariagiovanna Gianfreda

The inspiration for this theoretical paper comes from recent experiments on a PT-symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two PT transitions depending on the size of the coupling parameter \epsilon. For small \epsilon the PT symmetry is broken and the system is not in equilibrium, but when \epsilon becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large $$\epsilon$$ the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of $$\epsilon$$.

http://arxiv.org/abs/1305.7107
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)